Overview
This lecture covers the definitions and properties of even and odd trigonometric functions and how to use these properties to evaluate trig values for negative angles.
Even and Odd Trigonometric Functions
- Even functions satisfy f(-x) = f(x); cosine and secant are even.
- Odd functions satisfy f(-x) = -f(x); sine, cosecant, tangent, and cotangent are odd.
Application Examples
- To evaluate cos(-45°): use the even property, cos(-45°) = cos(45°) = √2/2.
- To evaluate sin(-π/3): use the odd property, sin(-π/3) = -sin(π/3) = -√3/2 (since sine is negative in quadrant IV).
- To evaluate tan(-3Ï€/4): use the odd property, tan(-3Ï€/4) = -tan(3Ï€/4) = 1 (since both x and y are negative in quadrant III).
- For cosecant(5Ï€/6): since cosecant is odd, csc(-5Ï€/6) = -csc(5Ï€/6) = -2 (using reciprocal and reference angles).
- For secant(-11π/6): since secant is even, sec(-11π/6) = sec(11π/6) = 2√3/3 (using cosine's positive value in quadrant IV).
- For tangent(π): tan(π) = y/x = 0/(-1) = 0; same for negative π, since both refer to the same point (-1, 0).
Key Terms & Definitions
- Even Function — Function where f(-x) = f(x) (e.g., cosine, secant).
- Odd Function — Function where f(-x) = -f(x) (e.g., sine, cosecant, tangent, cotangent).
- Reference Angle — The acute angle a given angle makes with the x-axis.
- Reciprocal Identity — csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x).
- Unit Circle — Circle of radius 1 centered at the origin, used for evaluating trig functions.
Action Items / Next Steps
- Practice evaluating trig functions for negative angles using even/odd properties.
- Review the unit circle and signs of trig functions in each quadrant.